License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.14
URN: urn:nbn:de:0030-drops-66370
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6637/
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Makarychev, Konstantin ; Makarychev, Yury ; Sviridenko, Maxim ; Ward, Justin

A Bi-Criteria Approximation Algorithm for k-Means

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Abstract

We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor.

We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.

BibTeX - Entry

@InProceedings{makarychev_et_al:LIPIcs:2016:6637,
  author =	{Konstantin Makarychev and Yury Makarychev and Maxim Sviridenko and Justin Ward},
  title =	{{A Bi-Criteria Approximation Algorithm for k-Means}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{14:1--14:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6637},
  URN =		{urn:nbn:de:0030-drops-66370},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.14},
  annote =	{Keywords: k-means clustering, bicriteria approximation algorithms, linear programming, local search}
}

Keywords: k-means clustering, bicriteria approximation algorithms, linear programming, local search
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)
Issue Date: 2016
Date of publication: 06.09.2016

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